Questions tagged [normalization]
The word normalization refers to a procedure transforming an algebraic variety (more generally, a scheme) to a normal one via a birational morphism. This tag should be used only for questions in algebraic geometry rather than ones in analysis, logic or probability.
53 questions
1 vote
0 answers
221 views
Is the closure of a normal subvariety normal?
Let $X$ be an affine, complex variety of dimension $n \ge 3$. Suppose that $X$ has only one singular point, say $x$. Let $Y \subset X$ be a codimension one, irreducible subvariety such that $Y \...
- 1,122
2 votes
0 answers
101 views
Normal form of boolean expressions linear/affine w.r.t. conjunction and disjunction
$\DeclareMathOperator\Bool{Bool}$I am interested in boolean expressions that are linear/affine in the following sense. Let $\Bool(X)$ be the free boolean algebra over the set $X$. We can consider the ...
- 521
3 votes
0 answers
179 views
When does weak normalization imply strong normalization?
Is there a possibility to get strong normalization for some kind of $\lambda$-calculus out of weak normalization with some other assumptions? For example: The term $(\lambda_y z)((\lambda_x xx)(\...
- 193
2 votes
1 answer
253 views
When is the singularity of a semi-normal variety a double point singularity
Let $X$ be a semi-normal projective variety and $p: \widetilde{X} \to X$ be the normalization. Suppose that $\widetilde{X}$ is smooth and there exists two smooth divisors $D_1, D_2 \subset \widetilde{...
- 1,665
4 votes
1 answer
154 views
Normal form of framed links under Kirby moves
It's well known that any oriented closed 3-manifold (topological or smooth) can be obtained by surgerizing along a (framed oriented) link $L$ in the 3-sphere $S^{3}$. Even better, Kirby found a ...
- 5,738
7 votes
1 answer
457 views
Criterion for the consistency of pure type systems
Pure type systems are characterized in an almost combinatorial way: a set of axioms $\star_i : \star_j$, and a set of triples $(\star_i, \star_j, \star_k)$ saying when the dependent product $\prod_{x :...
- 2,146
1 vote
0 answers
231 views
Moduli interpretation of normalization of moduli space
The question is about formal and rigid geometry, but I would be interested in an answer from an algebraic geometry point of view as well. Let $\mathfrak{X}$ be a formal moduli space (e.g., the formal ...
- 949
3 votes
0 answers
330 views
If the normalization is affine, is it affine? (if quasiaffine)
I was surprised to find out that, even if the normalization $X^\nu$ of a scheme $X$ is affine, $X$ may not be affine (remove the line $x=y$ from their example to make the source affine). In the ...
- 1,134
8 votes
0 answers
410 views
Example of torsion differential forms
I am looking for an example of a normal affine variety $V$ over a perfect field $k$ such that the differentials $\Omega_{V/k}$ are not torsion free. If normality is not required, an example is given ...
- 227
4 votes
0 answers
162 views
Embedded normalization
Let $S$ be an irreducible surface in a 3-dimensional variety $X$ (everything taking place over $\mathbb{C}$, say). By Hironaka's therorem, we know for sure that there is an embedded resolution of $S$, ...
- 61
2 votes
0 answers
606 views
Normalized Laplacian matrix versus walk Laplacian matrix (or normalized adjacency matrix versus walk adjacency matrix)
In graphs, found that two different normalization matrices exist for Laplacian and adiacency matrix. I will ask about the adjacency matrix (for the Laplacian matrix the questions are the same). The ...
- 121
2 votes
2 answers
437 views
Why does a complex linear normalization of a real algebraic surface inherit a real structure?
Could you recommend any references to (some of) the following very basic assertions in algebraic geometry? (It seems unreasonable to reprove them in a research paper.) (1) Let a surface $X$ in $\...
2 votes
0 answers
179 views
Normalization of affine curves in singular surfaces
Let $X$ be a normal, isolated surface singularity with $x_0 \in X$ the unique singularity. Let $C \subset X$ be a hyperplane section i.e., defined by a single equation. Denote by $n:\widetilde{C} \to ...
- 2,062
0 votes
0 answers
100 views
Bijective restriction of the normalization morphism
Let $X$ be an integral separated scheme of finite type over $\mathbb{C}$. Consider the normalization morphism $f:X'\rightarrow X$. Can we always find an affine open $U\subset X'$ such that $f|_U:U\...
6 votes
1 answer
339 views
Does ampleness descend along finite maps?
First, let me emphasize that for $X$ a not-necessarily proper variety, we say that a line bundle $L$ on $X$ is ample, if for some positive integer $n$, $L^{\otimes n}$ arises as $j^*O(1)$ for some (...
- 2,894
13 votes
1 answer
539 views
Is height preserved in a normalization?
Let $R$ be a domain and $\tilde R$ its integral closure in its fraction field: $R\subset \tilde R\subset Frac(R)$. Is it true that a prime ideal $ \tilde {\mathfrak p} \subset \tilde R$ and its ...
- 48.3k
3 votes
1 answer
233 views
$\widetilde{R}=\bigcap_{\mathsf{ht}(\mathfrak p)=1}R_\mathfrak p$
As we know every normal Noetherian domain $R$ can be written as $$R=\bigcap_{\mathsf{ht}(\mathfrak p)=1}R_\mathfrak p.$$ I'm asking myself the following question: Question: If the normalization of $\...
- 1,280
3 votes
1 answer
316 views
Ring of sections and normalization
Let $D$ be a base-point-free divisor on a normal projective variety $X$, and let $Y$ be the image of the morphism $f_{D}:X\rightarrow Y$ induced by $D$. Assume that $f_D$ is birational. Now, let $X(D)...
user114666
5 votes
0 answers
784 views
Picard group of normalization
Let $X$ be a projective variety with at worst (analytic) normal crossings singularities and $\pi:\tilde{X} \to X$ be the normalisation. Is there a "nice" description relating the picard group of $X$ ...
- 2,126
4 votes
0 answers
167 views
Positivity of the ramification divisor
Let $X$ be a non-normal surface such that $K_X$ is a pseudo-effective divisor and ${\rm Bs}_{-}(K_X)$ (the diminished base locus of $K_X$) equals, at least set-theoretically, the non-normal locus of $...
- 1,593
11 votes
3 answers
1k views
How to handle sums in Tait's reducibility proof of strong normalisation?
I've been reading Girard et al's 'Proofs and Types', which in Chapter 6 presents a proof of strong normalisation for the simply typed lambda calculus with products and base types. The proof is based ...
- 113
1 vote
0 answers
62 views
A question about the prediction error
I am reading about the prediction error estimation and I found the following: Suppose we have ${\mathbf{Y}}=\mathbf{x}_0+ \epsilon$, where, $\epsilon$ is normally distributed as $\sim \mathcal{N}(0, \...
- 11
3 votes
0 answers
333 views
Understanding Strong Normalization for Identity Types in Martin-Löf Intensional Type Theory [closed]
Roughly, the strong normalization property for Martin-Löf Intensional Type Theory (MITT) tells us that every closed term $t$ of type $M$ will eventually reach a canonical normal form $t’$ such that it ...
- 765
2 votes
1 answer
271 views
If $X$ has non-singular normalization $\dim (\mathrm{Sing(X)})=\dim (X)-1$?
Let $X\subseteq\mathbb{P}^{N}$ be a quasiprojective variety of dimension $N-1$, and let $$ \nu:X^{\nu}\rightarrow X $$ be its normalization. Let us suppose that $X^{\nu}(\neq X)$ is smooth. I wonder ...
- 113
3 votes
0 answers
563 views
the normalized blowup
Let $X$ be a normal variety over $\mathbb{C}$ and $x\in X$ a singular point. Let $f:Y^{\nu}\to X$ be the normalized blowup at $x\in X$. (i.e. $f$ is a composition of the blowup $Y:=Bl_xX\to X$ and ...
- 31
2 votes
2 answers
379 views
Normalization of a Noetherian local domain and line bundles on the punctured spectrum
Let $A$ be a Noetherian local domain ($2$-dimensional if needed) such that its punctured spectrum $U$ is regular, and let $A'$ be the normalization of $A$. 1) Is it possible for $A'$ to have ...
- 835
1 vote
0 answers
228 views
Twisting locally free sheaves in characteristic $p$
Let $X$ be an irreducible nodal projective curve over an algebraically closed field of characteristic $p>0$. Denote by $\pi:\tilde{X} \to X$ the normalization of $X$. Recall, the short exact ...
- 2,061
2 votes
1 answer
506 views
Projective normality of cones over projectively normal varieties
Let $X\subseteq\mathbb{P}^n$ be a smooth subvariety, with homogeneus ideal $I\subseteq k[x_0,\ldots,x_n]$. Let $C(X)\subseteq\mathbb{P}^{n+1}$ be the projective cone over $X$, so that $C(X)$ is ...
- 1,189
3 votes
1 answer
770 views
Can height one maximal ideals in the normalization contract to non-height one primes in the base?
Let $R$ be a local (Noetherian) integral domain of dimension greater than one. Can the integral closure (i.e. normalization) of $R$ have a maximal ideal of height one?
- 1,994
3 votes
1 answer
240 views
Eigenfunctions to 2nd-order Differential Operators: Relation between Frobenius Series Solution and Eigenfunction Normalised to the Delta Function
Consider the 2nd-order linear ODE $x f^{''}(x) + x (\beta - 2 \alpha x) \kappa / \sigma f^{'}(x) - 1 / \sigma \left[ 2 \alpha \kappa - \lambda^2 (\beta - 2 \alpha x)^2 \right] f(x) = 0$, where $\sigma&...
- 41
6 votes
1 answer
764 views
Normalization of a curve and push forward of vector bundles
Let $C$ be a projective curve (over an algebraically closed field, not necessarily of characteristic zero) which is smooth except for exact one node. Let $\pi:\tilde{C} \to C$ be its normalization. ...
- 833
10 votes
2 answers
940 views
Equivariant normalization?
Let $G=\mathrm{Gl}_n\mathbb C$ and let $X$ be an affine $G$-variety. Let $\phi:\tilde X\to X$ be the normalization of $X$, i.e. the spectrum of the integral closure of $\mathbb C[X]$ in its fraction ...
- 4,982
4 votes
1 answer
747 views
When is normalization functorial?
Let $X$ and $Y$ be two irreducible, affine $\newcommand{\C}{\mathbb C}\C$-varieties. Let $f:X\to Y$ be a morphism. Denote by $u:\tilde X\to X$ and $v:\tilde Y\to Y$ their normalizations. Now, if $f$ ...
- 4,982
3 votes
2 answers
3k views
meaning of normalization
I have seen the following construction and I would be very happy if someone could explain its meaning to me. We start from a smooth projective algebraic variety $X$ over a field of characteristic ...
- 33
7 votes
1 answer
2k views
Relation between blowup and normalization
Let $X$ be a variety over an algebraically closed field with null characteristic. Let $C$ be a smooth subvariety of $X$ of dimension 1, and let $x$ be a point of $C$. We assume that $X$ is ...
- 1,044
6 votes
3 answers
1k views
computational complexity of primitive recursive functions
If we have a rewrite system for primitive recursive functions, which simplifies each term according to how the function was defined, then what is the computational complexity of this calculation? That ...
- 63
0 votes
0 answers
297 views
L_2-norm representation
Let $$ f^{\alpha}_+(x)=\frac{1}{\Gamma(\alpha+1)}\sum_{k\ge 0}(-1)^k{\alpha+1 \choose k}(x-k)^{\alpha}_+, $$ where $\alpha > -\frac 12$. I am wondering if one can get nice representation of $L^2$-...
- 71
1 vote
1 answer
930 views
Is this function field extension a Galois extension ?
Setting and question Let $X$ be a variety over an algebraically closed field of null characteristic, and let $C$ be a (regular if you want) curve included in $X$. Consider $X'$ the normalization of $...
- 1,044
7 votes
1 answer
2k views
Noether normalization vs. normalization of varieties
As far as I can tell, Noether normalization uses the term "normalization" in the English sense, that something has been given a standard form. And as such it's not intrinsically related to ...
- 28.2k
4 votes
1 answer
805 views
Finiteness of normalization of Noetherian normal domain
I have the following question: Let $A$ be an integrally closed Noetherian domain, $K$ its field of fractions. let $L$ be a finite extension of $K$, and $B$ the integral closure of $A$ inside $L$. Is ...
- 5,622
3 votes
1 answer
581 views
Which monomial subalgebras are direct summands of polynomial rings
Let $S=k[x_1,\dots,x_n]$ be a polynomial ring, and $A:=k[x^{u^{(1)}}, \dots x^{u^{(l)}}]$ a monomial subalgebra, generated by monomials $x^{u^{(i)}} = \prod_{j=1}^n x_j^{u^{(i)}_{j}}$ with $u^{(i)} \...
- 2,001
3 votes
1 answer
253 views
Simple reference for valuative criterion of integrality?
I'd like to see a complete proof of the simplest version of the following rough statement: "If $f/g$ is a rational function on a reduced scheme ($g$ not a zero divisor), and $f/g$ doesn't have poles ...
- 28.2k
0 votes
0 answers
277 views
Does the normalization of a projective morphism determine the line bundle?
Let $X$ be a smooth, complete algebraic variety and suppose I have two projective, birational morphisms $$f:X \to \mathbb{P}^n$$ and $$g:X \to \mathbb{P}^m,$$ such that the image of $f$ is the ...
- 1
2 votes
1 answer
1k views
Line bundles, linear systems and normalization
One example that I always have in mind is that the plane nodal (or even the plane cuspidal) cubic curve $X$ is obtained by an appropirate 2-dim linear subsystem of $|\mathcal{O} (3)|$ on $\mathbb{P}^...
- 23
2 votes
1 answer
410 views
On the normalization and the quotient of the structure sheaves
Let $\nu:\tilde{X}\to X$ be the normalization of a projective variety with non-isolated singularity. The usual object to consider is $\nu_*\mathcal{O}_{\tilde{X}}/\mathcal{O}_X$. For example, one ...
- 2,282
19 votes
4 answers
4k views
Flatness of normalization
Let $X$ be a noetherian integral scheme and let $f \colon X' \to X$ be the normalization morphism. It is known that, if non trivial, $f$ is never flat (see Liu, example 4.3.5). What happens if we ...
- 3,764
1 vote
2 answers
563 views
Dimensionality of a map -- distance
Hello, I am looking for some words to describe what going on here. I'm sure this is not an original thought, so I'd like to read up on more from others who have thought out this topic further. FORMAT ...
- 113
7 votes
1 answer
644 views
Normality of a locus of points in projective space
Let $U_{d,n}\subseteq(\mathbb{P}^d)^n$ denote the locus of $n$-distinct points in projective space $\mathbb{P}^d$ that lie on a rational normal curve of degree $d$, and let $V_{d,n}$ denote its ...
- 1,871
8 votes
2 answers
475 views
Doing explicit computations with coordinate rings
Suppose that we are given an integral $k$-algebra $A$ of finite type explicitly, by which I mean that we are given the generators of the defining ideal $J$ where $A = k[x_1,...,x_n]/J$. What kinds of ...
- 3,988
3 votes
2 answers
614 views
Is weak normality stable under completion?
I'm curious if anyone knows a reference for the following. It seems like someone must have done this somewhere, but I couldn't find a reference. Recall that an excellent reduced noetherian ring $R$ ...
- 21k